Hunting submartingales in the jumping voter model and the biased annihilating branching process

Two species (designated by 0's and 1's) compete for territory on a lattice according to the rules of a voter model, except that the 0's jump do spaces and the 1's jump dl spaces. When d(0) = d(1) = 1 the model is the usual voter model. It is shown that in one dimension, if d(1) > d(0) and d(0) = 1, 2 and initially there are infinitely many blocks of i's of length greater than or equal to d(1), then the 1's eliminate the 0's. It is believed this may be true whenever d(1) > d(0). In the biased annihilating branching process particles place offspring on empty neighbouring sites at rate lambda and neighbouring pairs of particles coalesce at rate 1. In one dimension it is known to converge to the product measure density lambda/(1 + lambda) when lambda greater than or equal to 1/3, and the initial configuration is non-zero and finite. This result is extended to lambda greater than or equal to 0.0347. Bounds on the edge-speed are given.